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On Orthocentric Systems in Strictly Convex Normed Planes

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On Orthocentric Systems in Strictly Convex Normed Planes E extracta mathematicae Vol. 24, N¶um.1, 31 { 45 (2009) On Orthocentric Systems in Strictly Convex Normed Planes Horst Martini, Senlin Wu ¤ Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany [email protected] Department of Applied Mathematics, Harbin University of Science and Technology 150080 Harbin, China, senlin [email protected] Presented by Pier L. Papini Received November 12, 2008 Abstract: It has been shown that the three-circles theorem, which is also known as T»it»eica's or Johnson's theorem, can be extended to strictly convex normed planes, with various ap- plications. From this it follows that the notions of orthocenters and orthocentric systems in the Euclidean plane have natural analogues in strictly convex normed planes. In the present paper (which can be regarded as continuation of [5] and [14]) we derive several new charac- terizations of the Euclidean plane by studying geometric properties of orthocentric systems in strictly convex normed planes. All these results yield also geometric characterizations of inner product spaces among all real Banach spaces of dimension ¸ 2 having strictly convex unit balls. Key words: Birkho® orthogonality, Busemann angular bisector, C-orthocenter, inner product space, isosceles orthogonality, Minkowski plane, normed linear space, orthocenter, orthocen- tric system, three-circles theorem. AMS Subject Class. (2000): 46B20, 46C15, 52A10. 1. Introduction By (M2; C) we denote an arbitrary normed or Minkowski plane (i.e., a real two-dimensional normed linear space) with unit circle C, origin O, and norm k¢k. We restrict our discussion to strictly convex Minkowski planes, whose unit circles are strictly convex curves with midpoint O, thus not containing a non-degenerate segment. Basic references to the geometry of Minkowski planes and spaces are [18], [16], [17], and the monograph [19]. For any point x 2 (M2; C) and any number ¸ > 0, the set C(x; ¸) := x+¸C is said to be the circle centered at x and having radius ¸. For x 6= y, we denote by hx; yi the line passing through x and y, by [x; y] the segment between x and y, and by [x; yi the ray with starting point x passing through y. The ¤This work is partially supported by NSFC (China), grant 10671048. 31 32 h. martini, s. wu distance from a point x to a non-empty set A is denoted by d(x; A), i.e., d(x; A) = inffkz ¡ xk : z 2 Ag. It has been shown by E. Asplund and B. GrÄunbaum in [5] that the follow- ing theorem, which is the extension of the classical three-circles theorem in the Euclidean plane, also holds in strictly convex, smooth Minkowski planes. (The three-circles theorem is also called T»it»eica's or Johnson's theorem; see the survey [13] and the monograph [12, p. 75]. For the related concept of orthocentricity, even in Euclidean n-space with n ¸ 2, we refer to the survey contained in [9] and to [7].) Theorem 1.1. If three circles C(x1; ¸), C(x2; ¸), and C(x3; ¸) pass through a common point p4 and intersect pairwise in the points p1, p2, and p3, then there exists a circle C(x4; ¸) such that fp1; p2; p3g ⊆ C(x4; ¸). Actually, the smoothness condition can be relaxed, i.e., Theorem 1.1 holds in strictly convex Minkowski planes (cf. [19, Theorem 4.14, p. 104] and [14]). This theorem is also basic for extensions of Cli®ord's chain of theorems to strictly convex normed planes; see [15]. The point p4 in Theorem 1.1 is called the C-orthocenter of the triangle p1p2p3, and by Theorem 1.1 it is also evident that pi is the C-orthocenter of the triangle pjpkpl, where fi; j; k; lg = f1; 2; 3; 4g. Thus it makes sense to call a set of four points, each of which is the C-orthocenter of the triangle formed by the other three points, a C-orthocentric system. See [20] for another concept of orthocenters which is related to Birkho® orthogonality. The proof of the above theorem for strictly convex Minkowski planes is based on the following properties of strictly convex Minkowski planes (cf. [5] and [18, p. 106]), which will be used throughout the paper: Lemma 1.2. Let (M2; C) be a strictly convex Minkowski plane. If x1 6= x2 and fy1; y2g ⊆ C(x1; ¸) \C(x2; ¸), then x1 + x2 = y1 + y2. Lemma 1.3. Any three non-collinear points in a strictly convex Minkowski plane are contained in at most one circle. The following facts are well known in Euclidean geometry: 1: The three altitudes of a triangle intersect in a point called orthocenter of that triangle. on orthocentric systems 33 2: The altitude to the base of an isosceles triangle bisects the corresponding vertex angle. 3: If one of the altitudes of a triangle is also an angular bisector, then the triangle is isosceles. 4: The altitude to the base of an isosceles triangle bisects its base. 5: If one of the altitudes of a triangle is also a median (i.e., a segment from a vertex to the midpoint of the opposite side), then the triangle is isosceles. As the notion of C-orthocenter can be viewed as a natural extension of that of orthocenter in Euclidean geometry, one may ask whether results related to orthocenters in Euclidean geometry still hold in Minkowski geometry in the sense of C-orthocenters. It is our aim to continue the investigations from [14] in this spirit. For our discussion we need the de¯nitions of isosceles orthogonality, Birkho® orthogonality, and Busemann angular bisectors. Let x; y 2 (M2; C). The point x is said to be isosceles orthogonal to y if kx + yk = kx ¡ yk, and in this case we write x ?I y (cf. [10]). On the other hand, x is said to be Birkho® orthogonal to y if kx + tyk ¸ kxk holds for all t 2 R, and for this we write x ?B y (cf. [6]). We refer to [10], [11], and [2] for basic properties of isosceles orthogonality and Birkho® orthogonality, and to [4] and [3] for a detailed study of the relations between them. For non-collinear rays [p; ai and [p; bi, the ray · À 1³ a ¡ p b ¡ p ´ p; + + p 2 ka ¡ pk kb ¡ pk is called the Busemann angular¡ bisector of¢ the angle spanned by [p; ai and [p; bi, and it is denoted by AB [p; ai; [p; bi (cf. [8]). It is trivial to see that when ka ¡ pk = kb ¡ pk, then ¡ ¢ h 1 E A [p; ai; [p; bi = p; (a + b) : B 2 34 h. martini, s. wu 2. Some lemmas The following lemmas are needed for our investigations. Lemma 2.1. (cf. [1]) Let (M2; C) be a strictly convex Minkowski plane. Then, for any x 2 (M2; C)nfOg and any number ¸ > 0, there exists a point y 2 ¸C (unique except for the sign) such that x ?I y. Lemma 2.2. Let C(x1; ¸) and C(x2; ¸) be two circles in a strictly convex 0 0 Minkowski plane (M2; C). If fw; zg ⊆ C(x1; ¸), fw ; z g ⊆ C(x2; ¸), and w¡z = w0 ¡ z0, then 0 0 d(x1; hw; zi) = d(x2; hw ; z i): Proof. Without loss of generality, we can suppose that x1 = x2 = O and ¸ = 1. By the assumed strict convexity, kw ¡ zk = kw0 ¡ z0k = 2 implies that 0 0 0 0 w = ¡z and w = ¡z , yielding d(x1; hw; zi) = d(x2; hw ; z i) = 0. Now we consider the case when kw ¡ zk < 2. Again by strict convexity of (M2; C), any vector with norm < 2 is the sum of two unit vectors in a unique way (cf. [18, Proposition 14, p. 106]). Thus, either w = w0 and z = z0 or w = ¡z0 and 0 z = ¡w hold, and each of these two cases clearly gives that d(x1; hw; zi) = 0 0 d(x2; hw ; z i). Lemma 2.3. ([4, (4.12)]) If for any x; y 2 (M2; C) with x ?I y there exists a number 0 < t < 1 such that x ?I ty, then (M2; C) is Euclidean. The following lemma follows immediately from Lemma 2.3. Lemma 2.4. If for any x, y 2 (M2; C) with x ?I y there exists a number t > 1 such that x ?I ty, then (M2; C) is Euclidean. Lemma 2.5. (cf. [4, (10.2)]) A Minkowski plane (M2; C) is Euclidean if and only if the implication x ?B y ) x ?I y holds for any x, y 2 C. Lemma 2.6. (cf. [4, (10.9)]) A Minkowski plane (M2; C) is Euclidean if and only if the implication x ?I y ) x ?B y holds for any x, y 2 C. on orthocentric systems 35 Lemma 2.7. (cf. [14]) Let fp1; p2; p3; p4g be a C-orthocentric system in a strictly convex Minkowski plane (M2; C). If xi is the circumcenter of the triangle pjpkpl, where fi; j; k; lg = f1; 2; 3; 4g, then fx1; x2; x3; x4g is also a C-orthocentric system and pi ¡ pj = xj ¡ xi: Lemma 2.8. Let (M2; C) be a strictly convex Minkowski plane. For any x, y 2 (M2; C)nfOg with x ?I y, let p3 = y, p4 = ¡y, x1 = x, x2 = ¡x, and ¸ = kx + yk. Then there exist two points p1 2 C(x2; ¸) and p2 2 C(x1; ¸) such that fp1; p2; p3; p4g is a C-orthocentric system, and that one of the following conditions is satis¯ed: (1) kp3 ¡ p1k = kp3 ¡ p2k, and p3 and the line hp1; p2i are separated by the line l0, which is passing through p4 parallel to hp1; p2i, £ ¤ p1+p2 (2) p4 2 p3; , 2 ¡ ¢ (3) p3 and the line hp1; p2i are separated by l0, and p4 2 AB [p3; p1i; [p3; p2i , (4) p3 and the line hp1; p2i are separated by the line l0, and hp1; p2i is a common supporting line of the circles C(x2; ¸) and C(x1; ¸).
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